Vector definitions (HL)

Displacement vectors can be translated in the plane or space, while maintaining their magnitudes and directions. Meanwhile position vectors represent the vector from origin to a specific point in space.

Geometrically, they can be visualized as arrows.

Last edited: 2026-06-09 changed to proper vector notations and updated explanations

Definitions

A vector or vector quantity, $\bm{v}$ or $\vec{v}$, is associated with a magnitude (size) and a direction. Magnitudes are non-negative, and are represented by $\lvert\bm v\rvert$, $\lvert\vec v\rvert$, or simply $v$. In contrast, a scalar or scalar quantity, $v$, is only associated with a magnitude. Though, a scalar could be negative.

For example “1 m north” is a vector, where as “1 m” is a scalar.

$\bm i$, $\bm j$, $\bm k$ are the unit vectors for the $x$, $y$, and $z$-axes. The axes are perpendicular to each other and intersect at the origin $(0, 0, 0)$. For example “1 m north and 1 m east” is equivalent to ”$\sqrt 2$ m northeast”.

IB accepts the following representations of a vector.

The first two representations may appear on exams. The third is standard notation accepted on exams and will be commonly used in these notes in inline notations. IB does not accept $(2, 3, -4)$ as a vector.

The magnitude of this vector is $\lvert\bm v\rvert = \sqrt{2^2 + 3^2 + (-4)^2} = \sqrt{29}$.

Scalar multiplication means multiplying (or rescaling) a vector by a scalar. If the scalar is positive, the resulting direction is same as before; if it is negative, the vector reverses direction. Parallel vectors are ones in the same or opposite directions. Some non-IB sources also use "antiparallel" to mean specifically vectors in opposite directions.

The zero vector, $\bm{0}$, or $\vec{0}$, is the vector with zero magnitude and in any arbitrary direction.

The unit vector, $\bm{\hat{v}}$, is the vector in the direction of $\bm{v}$ and with magnitude $1$.

A vector goes from its tail to its head.

A position vector is a vector from the origin to a specific point in space. For example, the tail of $\overrightarrow{OP}$ is $O$, while the head is $P$. Also the point $A(1, -2, 3)$ is associated with the vector $\overrightarrow{OA} = \langle1, -2, 3\rangle$. Position vectors are fixed in space. For the rest of the vectors discussion, position vectors are synonymous with “points”.

A displacement vector is a difference between two position vectors, or between two displacement vectors. For instance, $\overrightarrow{AB} = \langle2, -1, 0\rangle$ is the displacement from $\overrightarrow{OA} = \langle-3, 2, 4\rangle$ to $\overrightarrow{OB} = \langle-1, 1, 4\rangle$.

Displacement vectors are more powerful and useful because they can be moved around. Common types of displacement vectors include

• a displacement vector representing a line segment
• a direction vector of a line
• a direction vector on a plane
• a normal vector to a plane

In certain cases, such as when the displacement vector is used as a direction vector of a line or a normal vector of a plane, only the direction is of interest.

Position vectors have fixed tails and heads, while displacement vectors have tails and heads that can be translated together.